Properties

Label 2-1078-1.1-c1-0-31
Degree $2$
Conductor $1078$
Sign $-1$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2.82·5-s − 8-s − 3·9-s − 2.82·10-s + 11-s − 5.65·13-s + 16-s − 2.82·17-s + 3·18-s − 8.48·19-s + 2.82·20-s − 22-s − 8·23-s + 3.00·25-s + 5.65·26-s − 6·29-s + 8.48·31-s − 32-s + 2.82·34-s − 3·36-s − 6·37-s + 8.48·38-s − 2.82·40-s + 8.48·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.26·5-s − 0.353·8-s − 9-s − 0.894·10-s + 0.301·11-s − 1.56·13-s + 0.250·16-s − 0.685·17-s + 0.707·18-s − 1.94·19-s + 0.632·20-s − 0.213·22-s − 1.66·23-s + 0.600·25-s + 1.10·26-s − 1.11·29-s + 1.52·31-s − 0.176·32-s + 0.485·34-s − 0.5·36-s − 0.986·37-s + 1.37·38-s − 0.447·40-s + 1.32·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + 8.48T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.530730185409156455610738588161, −8.734877997248979398883447111660, −8.039633612488981166741343372411, −6.85101474999731051432473448282, −6.17053049575869876704436871049, −5.42267652915547103492609807415, −4.18712210120360699910405087214, −2.46348138242626626833794123960, −2.08526057766359163412999320943, 0, 2.08526057766359163412999320943, 2.46348138242626626833794123960, 4.18712210120360699910405087214, 5.42267652915547103492609807415, 6.17053049575869876704436871049, 6.85101474999731051432473448282, 8.039633612488981166741343372411, 8.734877997248979398883447111660, 9.530730185409156455610738588161

Graph of the $Z$-function along the critical line